## Abstract

A life table is a table which shows, at each age, the probability that a person in a given population will die before their next birthday. It can be used to calculate life expectancy and healthy life expectancy for people of different ages. In this work, using longitudinal datasets and panel data methods, we produce life tables for different subgroups of the population, defined according to cohabitation status, employment and other factors. As a first step, we estimate the dynamics of factors which are of particular importance in people’s lives: health, labour market participation, cohabitation and mortality. The significance of these variables is twofold: they determine the well-being of individuals, but the variables also determine the resources available to the individuals in times of ill health. Using the British Household Panel Survey, we analyse the extent to which these variables are influenced by one another, and by exogenous factors such as education and ethnicity. Estimating a system of probit models using simulation techniques, we are able to distinguish the effects of the exogenous and endogenous variables from state dependence and unobserved heterogeneity. We also correct for attrition and the initial conditions problem. We estimate time trends in mortality, health and other dependent variables to investigate whether a compression of morbidity has occurred in the recent past. Finally, the parameter estimates are used to simulate life tables for various sub-groups in the population and compare measures of life expectancy and healthy life expectancy for different groups.

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## Notes

- 1.
\(\left\lceil x\right\rceil \) denotes the value of

*x*rounded upwards to the next integer. Hence, we order the columns and rows in the covariance matrix first according to year and then according to equation. - 2.
We provide an example of such a life table in Appendix “Example of a Life Table”.

- 3.
We do not report here how the first period error terms are derived. In order to obtain these, we use a simulation algorithm similar to that of the maximum likelihood procedure detailed in the main text. This way we take into account that the conditional distribution of

*v*depends on the starting value of the dependent variables of the model. - 4.
\(\left\lceil x\right\rceil \) denotes the value of

*x*rounded upwards to the next integer.

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## Additional information

This research was supported by a grant from the EPSRC.

## Appendices

### Appendix

### Simulating a Population Based on Model Estimates

The maximum likelihood procedure has provided us with parameter estimates for the econometric model. For simplicity, we partition these parameters into four groups: one is denoted *β*, one is denoted *d* the third one denoted *θ*, and the fourth one is denoted *κ*. The parameter vector *θ* contains all parameters related to the initial conditions problem, i.e. the parameters *θ*
_{0}, *θ*
_{1} and *θ*
_{2} from Eq. 7. Hence, this parameter vector relates to all variables which remain constant over the projection period. The parameter vector *d* contains all parameters related to state dependence. The parameter vector *β* contains all parameters related to time-varying exogenous variables such as age and time, and the parameter vector *κ* contains the parameters of the covariance matrix of the error terms (Σ; we have denoted these parameters *ρ*, *σ* and *ω* in the paper).

The different sets of parameters are outlined in Table 19 below.

Obviously, the first three sets of parameter vectors contain parameters for each of the four estimating equations—hence, we can define vectors *β*
_{
a
}, *d*
_{
a
} and *θ*
_{
a
} for the parameters of the survival equation, and similarly for the other three equations.

### Simulating a Subpopulation

We determine a sample size *N*, in this case 10,000. Assuming a maximum life length of *T* = 100 years from the start year (since we focus on 50-year olds, this is reasonable), we need to simulate a matrix of error terms and then make sure they have the appropriate correlations with each other (determined by Ω).

#### Simulating Error Terms

First, we simulate a matrix of standard normals:

where *u* is a 4*T*×*N* matrix with each element \(u_{ij}\backsim U\left( 0,1\right) \) and \(F^{-1}\left( \cdot \right) \) is the inverse of the cdf of a standard normal distribution. Obviously, *v* is also 4*T*×*N* and the observations are iid with mean zero and variance 1.^{Footnote 3}

Next, we build the 4*T*×4*T* covariance matrix Σ. This matrix is defined as

where^{Footnote 4}
\(a=\left\lceil \frac{i}{4}\right\rceil \) and \( b=\left\lceil \frac{j}{4}\right\rceil \) identify the corresponding estimating equation,

and

identify the corresponding years. Hence, the element Σ_{
ij
} tells us how the error term at time *t* in equation *a* is correlated with the error term at time *s* in equation *b*. The parameter *σ*
_{
ab
} is the covariance of the individual effect in equation *a* with the corresponding individual effect in equation *b*. Likewise, *ω*
_{
ab
} is the covariance of shocks in equation *a* with shocks in equation *b*. Whenever *a* = *b*, the corresponding variance is included.

Since Σ is positive definite and symmetric, we can carry out Cholesky decomposition. Hence, we define the 4*T*×4*T* matrix *L* as the lower diagonal Cholesky factor of the covariance matrix Σ:

Then, premultiplying the matrix of simulated error terms *v* by the Cholesky factor,

we get a *N*×4*T* matrix of error terms, distributed according to \( N\left( 0,\Sigma \right) \). Next, we partition the matrix *e* so that we get one *N*×*T* matrix for each estimating equation. We denote this matrices *e*
^{a}, *e*
^{w}, *e*
^{c} and *e*
^{h} for convenience. They are derived from the oringinal matrix according to the equation

and similarly for the four other equations.

#### Determining the Individual Effect

Next, we need to determine the individual effect. The deterministic part (see Eq. 7) is the same for all individuals in a subgroup. A subgroup is characterised by their values of the independent variables which remain constant over time as well as the initial state, represented by the variables *W*
_{0}, *C*
_{0} and *H*
_{0}. If we denote all other independent variables which remain constant (i.e. education, sex, ethnicity) *Z*, we can define the vector

Then, the deterministic part of the individual effect in equation *j* (\(j\in \left\{ a,w,c,h\right\} \)) can be defined as

where we have suppressed the random part of *α* from Eq. 7 since it is included in the matrix *e*. Also, since all individuals in a certain subgroup have the same individual effects *α*
_{
j
} (again, ignoring the random term), we suppressed the individual index *i* used in Eq. 7.

#### Simulating Outcomes

Having defined the individual effect *α*
_{
j
}, and constructed the matrix of error terms *e*, it is straightforward to simulate a population. This is done recursively, starting in year 1 and calculating the current state in all dimensions \(\left( A,W,C,H\right) \) for all simulated individuals, then moving on to the next year. Hence, we use the following procedure:

where *A*
_{
it
} takes on the value 1 if simulated individual *i* survives to period *t*; *X*
_{
t
} represents exogenous variables changing over time (age, time trend) and \(1\left[ \cdot \right] \) is the indicator function, taking on the value 1 whenever the expression in the square brackets is true.

For the other dependent variables, we follow the same procedure; also taking into account that individuals need to be alive to be working, cohabiting and healthy. Hence,

and then, for the remaining two, we also add simultaneously determined variables (such as *W*
_{
it
}); hence:

Hence, at the end of this exercise, we have four *N*×*T* matrices of simulated outcomes for the four variables *A*, *W*, *C* and *H*. This means that we can easily obtain life expectancy measures by simple matrix manipulations. For example:

Similarly, healthy life expectancy can be calculated as:

And the same goes for other combinations of the dependent variables, such as working healthy life expectancy etc.

### Analysing the Effects of Changing Status

Next, we want to analyse the effect of moving an individual from a certain starting state to another one, without changing other characteristics. Obviously, the difference in, say, life expectancy between two groups *i* and *j* (which we call ‘gap’ in the paper) is simply the difference between the two:

However, when we consider moving an individual from one state to another, we want to take into account the fact that they can be assumed to be different from individuals in the destination category—and this is arguably the reason why they were actually in a different category at the outset. This differences between individuals belonging to different groups are captured by the individual effect *α*
_{
i
} in our model. Hence, when we analyse the effect of moving an individual, we want to calculate counterfactual outcomes, based on a simulation where we keep *α*
_{
i
} constant but change the starting position in accordance with the destination category.

Consider the survival equation above. In this new setting, the ‘counterfactual’ survival in period 1 would be

where *α*
_{
a
} = *Gθ*
_{
a
} is determined according the *actual* starting position \(\left[ W_{0}\; C_{0}\; H_{0} \right] \) of the group we are considering, whereas the state dependence vector \(\left[ W_{0}^{\prime }\; C_{0}^{\prime }\; H_{0}^{\prime } \right] \) is determined by the *counterfactual* starting position of the destination group. The same procedure is used for the other four dependent variables.

Now, if we denote the life expectancy calculated according to this counterfactual experiment by *LE*
^{′}, we could calculate the expected gain from moving from one starting position to another one as *LE*
^{′} − *LE*. We have called this difference ‘gain’ in the paper. Obviously, this number can be larger or smaller than the difference Δ*LE* defined above.

### Example of a Life Table

Below we present two examples of the life tables which can be produced based on the parameter estimates from Section “Results”. In Table 20, we compare two males who are identical in all respects except for their educational attainment. In the left hand column, we present a life table for an individual with a university degree, and in the right column the corresponding table for someone without any education.

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### Cite this article

Karlsson, M., Mayhew, L. & Rickayzen, B. Individualised Life Tables.
*Population Ageing* **1, **153–191 (2008). https://doi.org/10.1007/s12062-009-9008-2

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### Keywords

- Disability
- Cohabitation
- Mortality
- Labour supply
- Maximum simulated likelihood
- Attrition